Empirical investigation of substrate systems allows determination of the thicknesses of, and associated optical property component values, (eg. refractive index (n) and extinction coefficient (k), or mathematically equivalent real (e1) and imaginary (e2) complex dielectric function values), for films thereon.
Optical property component values for substrate systems can be empirically obtained by many methods, such as those utilizing substrate system film effected changes caused in reflected and/or transmitted light beam intensities, and those utilizing substrate system film effected changes in ellipsometric PSI and DELTA values. However obtained, resulting optical property component values, when plotted as a function of the wavelength, (ie. photon energy, which comprises of a beam of electromagnetic radiation utilized in the investigation), typically present as a rather irregular plot, which irregular plot can not be accurately represented by a simple analytical mathematical function over the range of wavelengths, (ie. photon energies), typically of interest, (eg. zero (0.0) ev to six (6.0) ev, and above).
To understand the present invention, it is helpful to realize that the index of refraction of a substrate system film represents the ratio of the speed electromagnetic radiation traveling in vacuum to that in said film material, and that it is typically a complex number, (N=n+ik), where N is the Index of Refraction, n is termed the refractive index and k is termed the extinction coefficient. An alternative representation can be arrived at by the relation (e=N^2), and is termed the Dielectric Function, (ie. e=e1+ie2), where (e1=n^2−k^2) and (e2=2×n×k). It is also noted that the photon energy of a beam of electromagnetic radiation is related to the frequency thereof by the equation (E=h×F), where E is energy, (typically in electron volts ev), h is Plank's Constant, and F is Frequency. As well, Frequency, in electromagnetic radiation, is related to Wavelength by the relationship (C=F=W), where C is the speed of light and W is wavelength, hence (E=h×C/W). In addition, electrical properties of a material are related to the optical properties by known relationships. For instance, electrical conductivity is found to be proportional to e2.
One approach to modeling the Imaginary Part (e2) of a Dielectric Function is to identify some number of points thereupon and fit a polynomial thereto. Applying a Kramer-Kronig (K-K) Integration to the Imaginary Part (e2) provides the Real Part (e1). This is discussed in an Article titled “High Precision UV-Visible-Near-IR Stokes Vector Spectroscopy”, Zettler et al., Thin Solid Films, 234 (1993). It is also important to understand that the real and imaginary components of the Refractive Index (ie. n and k), are not independent of one another. As principal of causality applies, known as the Kramer-Kronig, (K-K), relationship, and theoretically, allows determining k(E) when n(E) is known, and vice-versa.
Continuing, other Dielectric Function Models have been developed for semiconductors. Adachi has developed a (K-K) consistent model suitable for below energy band-gap calculations that has been used to describe ternary alloys by interpolating parameters between binary endpoints, (see S. Adachi, “Optical dispersion relations for GaP, GaAs, GaSb, InAs InSb, Al(x)Ga(1−x)As, and In(1−x)As(y)P(1−y)”, Appl. Phys., Vol 66, p. 6030, 1989. The ability to interpolate parameters produces much more physically realistic dielectric models than simply averaging dielectric functions of the end-point binary materials. For terniary alloy models, the strong Critical Point (CP) structure of semiconductors causes simple averaging schemes to produce doubled (CP) structures at the energies of binary endpoints, (see P. G. Snyder, J. A. Woollam, S. A. Alterovitz and B. Johs, “Modeling Al(x)Ga(1−x)As Optical Constants as Functions of Composition”, J. Appl. Phys., Vol. 68, p. 5925, 1990). Forouhi and Bloomer have provided a dielectric model for semiconductors which is (K-K) consistent and has a very small set of coefficients, (see A. R. Forouhi and I. Bloomer, “Optical Properties of Crystaline Semiconductors and Dielectrics”, Phys. Rev. B, vol. 38, p. 1865, 1988 and U.S. Pat. No. 4,905,170). The Forouhi and Bloomer model, however, has been found to have insufficient flexibility to fit existing dielectric functions accurately enough for ellipsometric modeling. Oscillator ensembles, (eg. harmonic and Lorentz), have been used to describe the above energy-gap behavior of some semiconductors and the AlGaAs alloy system, (see a paper by M. Erman, J. B. Theeten, P. Chambon, S. M. Kelso and D. E. Aspnes, titled “Optical Properties and Damage Analysis of GaAs Single Crystals partly Amorphised by ion Implantation”, J. Appl. Phys. vol. 56, p. 2664, 1984; and a paper by H. D. Yao, P. G. Snyder and J. A. Woollam, titled “Temperature Dependence of Optical Properties of GaAs”, J. Appl. Phys., vol. 70, p. 3261, 1991; and a paper by F. Terry Jr., “A Modified Harmonic Oscillator Approximation Scheme for the Dielectric Constants of Al(x)Ga(1−x)As”, J. Appl., vol 70, p. 409, 1991). These models have been used to fit measured ellipsometric data, however, they are incapable of describing direct-energy-band-gap spectral regions, and they require extra fictitious oscillators to fill in the absorbtion between (CP's). Kim and Garland et al. have developed a (K-K) consistent model that can adequately describe a semiconductor dielectric function above, below and through the fundamental direct energy-gap, and this model has been applied to the AlGaAs alloy system, (see C. C. Kim, J. W. Garland, H. Abad and P. M. Raccah, “Modeling the Optical Dielectric Function of Semiconductors: Extension of the Critical-Point-Parabolic-Band Approximation”, Phys Rev. B, vol 45, p. 11749, 1982; C. C. Kim, J. W. Garland, H. Abad and P. M. Raccah, “Modeling the Optical Dielectric Function of the Alloy System Al(x)Ga(1−x)As”, Phys. Rev. B, vol. 47, p. 1876, 1993). This model can accurately describe the dielectric function and higher order derivatives. However, to determine required internal coefficients a two stage fitting process is used. First (CP) energies and broadening are determined by fitting derivatives of the dielectric function, and then the remaining internal coefficients are determined with the energies and broadenings fixed. It will be appreciated that this model then requires that the dielectric functions exist before the model can be fitted. Furthermore, attempts at fitting all coefficients simultaneously (as necessary for direct ellipsometer data fitting), are unlikely to succeed because of the highly correlated nature of the functions internal to the model. Over part of the spectral range, the modeled imaginary part of the dielectric function results from the difference of internal function values one-hundred (100) times larger than the final value. The internal coefficients are delicately balanced to produce the proper output, and small changes in (CP) energies can cause large deviations in the model output. In addition, Lorentzian Broadening, which is known to be wrong for elements and compounds, is utilized in by Kim & Garland et al. work. While the Kim & Garland et al. work is very interesting, there remained need for an improved Parametric Model such a that taught in U.S. Pat. No. 5,796,983, which is discussed in detail in the Disclosure and Detailed Description Sections of this Specification, along with other applicable Oscillator Structures, (eg. Gaussian, Narrow Lorentzian, Harmonic, TOLO, Ionic1, Ionic2). As demonstrated in the Detailed Description Section of this Specification, typical conventional practice is to place Oscillator Structures in a Plot of the Imaginary Part of the Dielectric Function (e2) such that at each wavelength the sum of the contributions results in said Imaginary Part. Typically this requires placing Oscillator Structures under peaks, and elsewhere, which placement is an acquired art.
Need remains for a simple mechanical approach to placing Oscillator Structures in a Plot of the Imaginary Part of the Dielectric Function (e2) such that summation of contributions thereof at each wavelength results in said Imaginary Part of the Dielectric Function (e2). Said approach should utilize (K-K) consistant Oscillator Structures so that the Real part of the Dielectric Function results from an integration procedure.